Banach空间中伪压缩映象不动点的迭代逼近

Let K be a nonempty closed convex subset of a real p-uniformly convex Banach space E and T be a Lipschitz pseudocontractive self-mapping of K with F(T) := {x ∈K : Tx=x}≠φ.Let a sequence {xn} be generated from x1 ∈K by xn 1 = αnxn bnTyn сnun,yn=a'nxn b'nTxn c'nun for all integers n≥1.Then ‖-Txn‖→0 as n→∞.Moreover,if T is completely continuous,then {xn}converges strongly to a fixed point of T.     

Convergence analysis of iterative sequences for a pair of mappings in Banach spaces

Let C be a nonempty closed convex subset of a real Banach space E. Let S ： C→ C be a quasi-nonexpansive mapping, let T ： C→C be an asymptotically demicontractive and uniformly Lipschitzian mapping, and let F ：= {x ∈C ： Sx = x and Tx = x}≠Ф Let {xn}n≥0 be the sequence generated irom an arbitrary x0∈Cby xn＋i=（1-cn）Sxn＋cnT^nxn, n≥0.We prove the necessary and sufficient conditions for the strong convergence of the iterative sequence {xn} to an element of F. These extend and improve the recent results of Moore and Nnoli.     

Banach空间中有限族渐进伪压缩映象的迭代程序

Let E be a real Banach space and K be a nonempty closed convex and bounded subset of E. Let Ti ： K→ K, i=1, 2,... ,N, be N uniformly L-Lipschitzian, uniformly asymptotically regular with sequences {ε^（i）n} and asymptotically pseudocontractive mappings with sequences {κ^（i）n}, where {κ^（i）n} and {ε^（i）n}, i = 1, 2,... ,N, satisfy certain mild conditions. Let a sequence {xn} be generated from x1 ∈ K by zn：= （1-μn）xn＋μnT^nnxn, xn＋1 ：= λnθnx1＋ [1 - λn（1 ＋ θn）]xn ＋ λnT^nnzn for all integer n ≥ 1, where Tn = Tn（mod N）, and {λn}, {θn} and {μn} are three real sequences in [0, 1] satisfying appropriate conditions. Then ｜｜xn- Tixn｜｜→ 0 as n→∞ for each l ∈ {1, 2,..., N}. The results presented in this paper generalize and improve the corresponding results of Chidume and Zegeye, Reinermann, Rhoades and Schu.     

Fixed Point Theorem of {a,b,c} Contraction and Nonexpansive Type Mappings in Weakly Cauchy Normed Spaces

Let C be a closed convex weakly Cauchy subset of a normed space X. Then we define a new {a,b,c} type nonexpansive and {a,b,c} type contraction mapping T from C into C. These types of mappings will be denoted respectively by {a,b,c}-ntype and {a,b,c}-ctype. We proved the following： 1. If T is {a,b,c}-ntype mapping, then inf{ ｜｜ T（x）-x｜｜ ：x C C} =0, accordingly T has a unique fixed point. Moreover, any sequence {Xn}n∈NN in C with limn→∞｜｜T（xn） - Xn｜｜ = 0 has a subsequence strongly convergent to the unique fixed point of T. 2. If T is {a,b,c}-ctype mapping, then T has a unique fixed point. Moreover, for any x∈C the sequence of iterates {Tn （x）}n∈N has subsequence strongly convergent to the unique fixed point of T. This paper extends and generalizes some of the results given in [2,4, 7] and [13].     

Modified mann iterations for nonexpansive semigroups in Banach space

Let E be a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from E to E^＊, and C be a nonempty closed convex subset of E. Let {T（t） ： t ≥ 0} be a nonexpansive semigroup on C such that F ：=∩t≥0 Fix（T（t）） ≠ 0, and f ： C → C be a fixed contractive mapping. If {αn}, {βn}, {an}, {bn}, {tn} satisfy certain appropriate conditions, then we suggest and analyze the two modified iterative processes as：{yn=αnxn＋（1-αn）T（tn）xn,xn=βnf（xn）＋（1-βn）yn
{u0∈C,vn=anun＋（1-an）T（tn）un,un＋1=bnf（un）＋（1-bn）vn
We prove that the approximate solutions obtained from these methods converge strongly to q ∈∩t≥0 Fix（T（t））, which is a unique solution in F to the following variational inequality：
〈（I-f）q,j（q-u）〉≤0 u∈F Our results extend and improve the corresponding ones of Suzuki [Proc. Amer. Math. Soc., 131, 2133-2136 （2002）], and Kim and XU [Nonlear Analysis, 61, 51-60 （2005）] and Chen and He [Appl. Math. Lett., 20, 751-757 （2007）].     

Ergodic Retraction Theorem and Weak Convergence Theorem for Reversible Semigroups of Non-Lipschitzian Mappings

Let G be a semitopological semigroup. Let C be a closed convex subset of a uniformly convex Banaeh space E with a Frechet differentiable norm, and T = {Tt ： t ∈ G} be a continuous representation of G as nearly asymptotically nonexpansive type mappings of C into itself such that the common fixed point set F（T） of T in C is nonempty. It is shown that if G is right reversible, then for each almost-orbit u（.） of T, ∩s∈G ^-CO{u（t） ： t ≥ s} ∩ F（T） consists of at most one point. Furthermore, ∩s∈G ^-CO{Ttx ： t ≥ s} ∩ F（T） is nonempty for each x ∈ C if and only if there exists a nonlinear ergodic retraction P of C onto F（T） such that PTs - TsP = P for all s ∈ G and Px ∈^-CO{Ttx ： s ∈ G} for each x ∈ C. This result is applied to study the problem of weak convergence of the net {u（t） ： t ∈ G} to a common fixed point of T.     

三步投影算法的收敛性及其在变分不等式组中的应用

First a general model for a three-step projection method is introduced, and second it has been applied to the approximation solvability of a system of nonlinear variational inequality problems in a Hilbert space setting. Let H be a real Hilbert space and K be a nonempty closed convex subset of H. For arbitrarily chosen initial points x0, y0, z0 ∈ K,compute sequences xn, yxn, zxn such that { xn+1=(1-αn-rn)xn+αxPk[yn-ρTyn]+rnun,yn=(1-β-δn)xn+βnPk[zn-ηTxn]+δnun,zn=(1-an-λn)xn+akPk[xn-γTxn]+λnwn.For η, ρ,γ>0 are constants,{αn}, {βn}, {an}, {rn}, {δn}, {λn} C [0,1], {un}, {vn}, {wn} are sequences in K, and 0≤n + rn ≤ 1,0 ≤βn + δn ≤ 1,0 ≤ an + λn ≤ 1,(A)n ≥ 0, where T : K → H is a nonlinear mapping onto K. At last three-step models are applied to some variational inequality problems.     

Weak Convergence Theorems for Nonself Mappings

Let E be a real uniformly convex and smooth Banach space, and K be a nonempty closed convex subset of E with P as a sunny nonexpansive retraction. Let T1, T2 : K → E be two weakly inward nonself asymptotically nonexpansive mappings with respect to P with a sequence {k(i)n}  [1, ∞)(i = 1, 2), and F := F(T1)∩F(T2) = . An iterative sequence for approximation common fixed points of the two nonself asymptotically nonexpansive mappings is discussed. If E has also a Fr′echet differentiable norm or its dual E*has Kadec-Klee property, then weak convergence theorems are obtained.     

Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces

This paper studies the convergence of the sequence defined by x0∈C,xn 1=αnu (1-αn)Txn,n=0,1,2,…, where 0 ≤αn ≤ 1, limn→∞αn = 0, ∑∞n=0 αn = ∞, and T is a nonexpansive mapping from a nonempty closed convex subset C of a Banach space X into itself. The iterative sequence {xn} converges strongly to a fixed point of T in the case when X is a uniformly convex Banach space with a uniformly Gateaux differentiable norm or a uniformly smooth Banach space only. The results presented in this paper extend and improve some recent results.     

空间中渐近非扩张型映射的渐近行为

Let X be a uniformly convex Banach space X such that its dual X^＊ has the KK property. Let C be a nonempty bounded closed convex subset of X and G be a directed system. Let ={Tt ： t ∈ G} be a family of asymptotically nonexpansive type mappings on C. In this paper, we investigate the asymptotic behavior of {Ttx0 ： t∈ G} and give its weak convergence theorem.     

关于伪压缩映象不动点迭代的一点注记

设$K$是自反的并且具有一致Gateaux可微范数的Banach空间$E$的非空有界闭凸子集.设$T:K\rightarrow K$是一致连续的伪压缩映象.假设$K$的每一非空有界闭凸子集对非扩张映象具有不动点性质.设$\{\lambda_n\}$是$(0,\frac{1}{2}]$中序列满足: (i) $\lim_{n\rightarrow \infty}\lambda_n=0$; (ii) $\sum_{n=0}^{\infty}\lambda_n=\infty$.任给$x_1\in K$,定义迭代序列$\{x_n\}$为:$x_{n+1}=(1-\lambda_n)x_n+\lambda_nTx_n-\lambda_n(x_n-x_1),n\geq 1.$若$\lim_{n\rightarrow \infty}\|x_n-Tx_n\|=0$, 则上述迭代产生的$\{x_n\}$强收敛到$T$的不动点.     

具误差隐格式迭代逼近严格伪压缩映像族公共不动点 (英)

设$K$是实Banach空间$E$中非空闭凸集, $\{T_i\}_i=1^{N}$是$N$个具公共不动点集$F$的严格伪压缩映像, $\{\alpha_n\}\subset [0,1]$是实数列, $\{u_n\}\subset K$是序列, 且满足下面条件 (i)\ 设$K$是实Banach空间$E$中非空闭凸集, $\{T_i\}_i=1^{N}$是$N$个具公共不动点集$F$的严格伪压缩映像, $\{\alpha_n\}\subset [0,1]$是实数列, $\{u_n\}\subset K$是序列, 且满足下面条件 (i)\ 设$K$是实Banach空间$E$中非空闭凸集, $\{T_i\}_i=1^{N}$是$N$个具公共不动点集$F$的严格伪压缩映像, $\{\alpha_n\}\subset [0,1]$是实数列, $\{u_n\}\subset K$是序列, 且满足下面条件 (i)\ 设K是实Banach空间E中非空闭凸集，{Ti}i=1^N是N个具公共不动点集F的严格伪压缩映像，{αn}包括于[0,1]是实数例，{un}包括于K是序列，且满足下面条件（i）0〈α≤αn≤1;（ii）∑n=1∞（1-αn）=＋∞.（iii）∑n=1∞ ‖un‖〈＋∞.设x0∈K,{xn}由正式定义xn=αnxn-1＋（1-αn）Tnxn＋un-1,n≥1,其中Tn=Tnmodn，则下面结论（i）limn→∞‖xn-p‖存在，对所有p∈F；（ii）limn→∞d（xn,F）存在，当d（xn,F）=infp∈F‖xn-p‖；（iii）lim infn→∞‖xn-Tnxn‖=0.文中另一个结果是，如果{xn}包括于[1-2^-n,1]，则{xn}收敛，文中结果改进与扩展了Osilike（2004）最近的结果，证明方法也不同。     

自反Banach空间中非扩张非自映射的粘滞迭代逼近方法

主要在自反和严格凸的且具有一致G(a)teaux可微范数的Banach空间中研究了非扩张非自映射的粘滞迭代逼近过程,证明了此映射的隐格式与显格式粘滞迭代序列均强收敛到它的某个不动点.     

关于渐近非扩张映象不动点迭代的一点注记

设E是一致凸Banach空间,C是E的非空闭凸子集,T:C→C是具有不动点的渐近非扩张映象．该文证明了在某些适当的条件下,由下列修改了的Ishikawa迭代程序所定义的序列{xn}=xn 1=rpn,pn=(1-an)xn anTmn ryn un,yn=(1-bn)xn bnTkn xn vn, (n≥1)弱收敛到t的不动点．     

增生算子粘性逼近的强收敛定理

假设$E$为实Banach空间, $A$为具有零点的增生算子. 定义序列 $\{x_n\}$如下: $x_{n+1}=\alpha_n f(x_n)+(1-\alpha_n)J_{r_n}x_n$, 这里$\{\alpha_n\}$, $\{r_n\}$ 满足一定条件的序列, 令$J_r=(I+rA)^{-1}$, $r>1$. 假如空间$E$有弱连续对偶映像,或者$E$为一致光滑的,均得到了序列 $\{x_n\}$的强收敛性结果.     

关于广义{\Phi}-增生算子的最速下降法的收敛定理

设$E$为一致光滑Banach空间,$A:E\to E$为有界次连续广义${\it \Phi} $-增生算子满足:对任意$x_0\in E$,选取$m\ge 1$,使得$\| x_0 - x^* \| \le m$且$\mathop {\underline {\lim } }\limits_{r \to \infty } {\it \Phi} (r) > m\left\| {Ax_0 } \right\|$.设$\{C_n\}$为$[0,1]$中数列满足控制条件: i)$C_n\to 0\,(n\to\infty)$; ii)$\sum\limits_{n = 0}^\infty {C_n } = \infty $.设$\{x_n\}_{n\ge0}$由下式产生x_{n + 1} = x_n - C_n Ax_n ,\q n \ge 0, \eqno{(@)}$$则存在常数$a>0$,当$C_n < a$时,$\{x_n\}$强收敛于$A$的唯一零点$x^{*}$.     

Fixed Point Theorem of $\{a,b,c\}$ Contraction and Nonexpansive Type Mappings in Weakly Cauchy Normed Spaces

Let $C$ be a closed convex weakly Cauchy subset of a normed space $X$. Then we define a new $\{a,b,c\}$ type nonexpansive and $\{a,b,c\}$ type contraction mapping $T$ from $C$ into $C$. These types of mappings will be denoted respectively by $\{a,b,c\}$-$n$type and $\{a,b,c\}$-$c$type. We proved the following:1. If $T$ is $\{a,b,c\}$-$n$type mapping, then $\inf\{\|T(x)-x\|:x\in C\}=0$, accordingly $T$ has a unique fixed point. Moreover, any sequence $\{x_{n}\}_{n\in \mathcal{N}}$ in $C$ with $\lim_{n\to \infty}\|T(x_{n})-x_{n}\|=0$ has a subsequence strongly convergent to the unique fixed point of $T$.2. If $T$ is $\{a,b,c\}$-$c$type mapping, then $T$ has a unique fixed point. Moreover, for any $x\in C$ the sequence of iterates $\{T^{n}(x)\}_{n\in \mathcal{N}}$ has subsequence strongly convergent to the unique fixed point of $T$.This paper extends and generalizes some of the results given in [2,4,7] and [13].     

涉及无限族非扩张映象$\{T_n\}_{n=1}^\infty$的迭代算法$x_{n 1}=\alpha_{n 1}f(x_n) (1-\alpha_{n 1})T_{n 1}x_n$

Under the framework of uniformly smooth Banach spaces, Chang proved in 2006 that the sequence {xn} generated by the iteration xn＋1 =αn＋1f（xn） ＋ （1 - αn＋1）Tn＋1xn converges strongly to a common fixed point of a finite family of nonexpansive maps {Tn}, where f ： C → C is a contraction. However, in this paper, the author considers the iteration in more general case that {Tn} is an infinite family of nonexpansive maps, and proves that Chang＇s result holds still in the setting of reflexive Banach spaces with the weakly sequentially continuous duality mapping.     

Banach空间中有限个增生算子公共零点的带误差项的迭代逼近

令E为实一致凸Banach空间,满足Opial条件或其范数是Frechet可微的.令为增生算子,满足值域条件且为非空闭凸子集且满足 .将引入新的带误差项的迭代算法并证明迭代序列弱收敛于{Ai}ki=1的公共零点.